Interpret all statistics for 1-Sample Sign

Find definitions and interpretation guidance for every statistic that is provided with the 1-sample sign analysis.

N

The sample size (N) is the total number of observations in the sample.

Interpretation

The sample size affects the confidence interval and the power of the test.

Usually, a larger sample size results in a narrower confidence interval. A larger sample size also gives the test more power to detect a difference. For more information, go to What is power?.

Median

The median is the midpoint of the data set. This midpoint value is the point at which half the observations are above the value and half the observations are below the value. The median is determined by ranking the observations and finding the observation that are at the number [N + 1] / 2 in the ranked order. If the number of observations are even, then the median is the average value of the observations that are ranked at numbers N / 2 and [N / 2] + 1.

Interpretation

The median of the sample data is an estimate of the population median.

Because the median is based on sample data and not on the entire population, it is unlikely that the sample median equals the population median. To better estimate the population median, use the confidence interval.

Note

To get both the confidence interval and the test results you must perform the analysis twice because Minitab only calculates one item at a time.

Confidence interval (CI)

The confidence interval provides a range of likely values for the population median. Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeated your sample many times, a certain percentage of the resulting confidence intervals or bounds would contain the unknown population median. The percentage of these confidence intervals or bounds that contain the median is the confidence level of the interval. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population median.

An upper bound defines a value that the population median is likely to be less than. A lower bound defines a value that the population median is likely to be greater than.

The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size.

The 1-sample sign test does not always achieve the confidence level that you specify because the sign test statistic is discrete. Because of this, Minitab calculates 3 confidence intervals with varying levels of precision. You should use the shortest interval for which the achieved confidence level is closest to the target confidence level.
  • The first confidence interval has the highest achievable confidence level that is less than the confidence level that you specify. The position indicates which observation Minitab uses for the upper and lower bound. or example, if the position is (7,14), the confidence interval is between the 7th smallest observation and the 14th smallest observation.
  • The second confidence interval is always at the confidence level that you specify. The upper and lower bounds of the confidence interval are not actual observations from the sample, so there is no position. Minitab uses nonlinear interpolation (NLI) to calculate this confidence interval.
  • The third confidence interval has the achievable confidence level that is greater than the confidence level that you specify. This is usually the widest interval.
Note

To get both the confidence interval and the test results you must perform the analysis twice because Minitab only calculates one item at a time.

Descriptive Statistics Sample N Median %Chromium 12 17.7
95% Confidence Interval for η Achieved Sample CI for η Confidence Position %Chromium ( 17.5, 18.1) 85.40% (4, 9) (17.4263, 18.7632) 95.00% Interpolation ( 17.4, 19) 96.14% (3, 10)

In these results, the estimate of the population median for the percentage of chromium is 17.7. You can use the second interval because it is the shortest interval that has a confidence interval closest to the target of 95%. You can be 95% confident that the population median is between 17.43 and 18.76.

Achieved Confidence

The achieved confidence level is the confidence level that is below or above the confidence level that you specify. The achieved confidence level indicates how likely it is that the population median is contained in the confidence interval. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population median.

The 1-sample sign test does not always achieve the confidence level that you specify because the sign test statistic is discrete. Because of this, Minitab calculates 3 confidence intervals with varying levels of precision. You should use the shortest interval for which the achieved confidence level is closest to the target confidence level.
  • The first confidence interval has the highest achievable confidence level that is less than the confidence level that you specify. The position indicates which observation Minitab uses for the upper and lower bound. or example, if the position is (7,14), the confidence interval is between the 7th smallest observation and the 14th smallest observation.
  • The second confidence interval is always at the confidence level that you specify. The upper and lower bounds of the confidence interval are not actual observations from the sample, so there is no position. Minitab uses nonlinear interpolation (NLI) to calculate this confidence interval.
  • The third confidence interval has the achievable confidence level that is greater than the confidence level that you specify. This is usually the widest interval.

Position

The position is the ordered rank of the data. The position indicates which observation Minitab uses for the upper and lower bound of the first and third confidence intervals. For example, if the position is (7,14), the confidence interval is between the 7th smallest observation and the 14th smallest observation.

For the second interval, Minitab uses nonlinear interpolation, which does not require a position.

Null hypothesis and alternative hypothesis

The null and alternative hypotheses are two mutually exclusive statements about a population. A hypothesis test uses sample data to determine whether to reject the null hypothesis.
Null hypothesis
The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value. The null hypothesis is often an initial claim that is based on previous analyses or specialized knowledge.
Alternative hypothesis
The alternative hypothesis states that a population parameter is smaller, larger, or different from the hypothesized value in the null hypothesis. The alternative hypothesis is what you might believe to be true or hope to prove true.

In the output, the null and alternative hypotheses help you to verify that you entered the correct value for the test median.

Number <

This value is the number of values in the sample that are less than the test median.

Interpretation

Minitab uses the number of values in the sample that are less than, equal to, and greater than the test median to calculate the p-value. Usually, larger differences between the number of observations that are greater or and less than the median produce lower p-values. Minitab removes the observations that are equal to the test median and then Minitab reduces the number of observations that it uses to calculate the p-value by the number of observations that it removed.

Number =

This value is the number of values in the sample that are equal to the test median.

Interpretation

Minitab uses the number of values in the sample that are less than, equal to, and greater than the test median to calculate the p-value. Usually, larger differences between the number of observations that are greater or and less than the median produce lower p-values. Minitab removes the observations that are equal to the test median and then Minitab reduces the number of observations that it uses to calculate the p-value by the number of observations that it removed.

Number >

This value is the number of values in the sample that are greater than the test median.

Interpretation

Minitab uses the number of values in the sample that are less than, equal to, and greater than the test median to calculate the p-value. Usually, larger differences between the number of observations that are greater or and less than the median produce lower p-values. Minitab removes the observations that are equal to the test median and then Minitab reduces the number of observations that it uses to calculate the p-value by the number of observations that it removed.

P-Value

The p-value is a probability that measures the evidence against the null hypothesis. A smaller p-value provides stronger evidence against the null hypothesis.

Interpretation

Use the p-value to determine whether the population median is statistically different from the hypothesized median.

To determine whether the difference between the population median and the hypothesized median is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
P-value ≤ α: The difference between the medians is significantly different (Reject H0)
If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis. You can conclude that the difference between the population median and the hypothesized median is statistically significant. Use your specialized knowledge to determine whether the difference is practically significant. For more information, go to Statistical and practical significance.
P-value > α: The difference between the medians is not significantly different (Fail to reject H0)
If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. You do not have enough evidence to conclude that the population median is significantly different from the hypothesized median. You should make sure that your test has enough power to detect a difference that is practically significant.
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